cp's OEIS Frontend

A057059 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ... Define i(m) and j(m) by R(i(m),j(m)) = m. Then a(n) = j(A057027(n)).

%I A057059 #74 Jan 06 2025 04:18:38
%S A057059 1,2,1,3,1,2,4,1,3,2,5,1,4,2,3,6,1,5,2,4,3,7,1,6,2,5,3,4,8,1,7,2,6,3,
%T A057059 5,4,9,1,8,2,7,3,6,4,5,10,1,9,2,8,3,7,4,6,5,11,1,10,2,9,3,8,4,7,5,6,
%U A057059 12,1,11,2,10,3,9,4,8,5,7,6,13,1,12,2,11,3,10
%N A057059 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ... Define i(m) and j(m) by R(i(m),j(m)) = m. Then a(n) = j(A057027(n)).
%C A057059 Since A057027 is a permutation of the natural numbers, every natural number occurs in this sequence infinitely many times.
%C A057059 Triangle of spiral permutations. In the Saclolo reference sigma_n(x) is called a spiral permutation. - _Michael Somos_, Apr 21 2011
%C A057059 Second inverse function (numbers of columns) for pairing function A194982. - _Boris Putievskiy_, Jan 10 2013
%C A057059 The triangle T(n, k) (see the formula by Michael Somos) has in row n a certain permutation of [1, 2, ..., n]. This permutation is useful for the proof of the identity Product_{k=1..n} f(sin(Pi*k/(2*n+1))) = Product_{m=1..n} f(sin(2*Pi*m/(2*n+1))) for any function f, n >= 1 (also for  n = 0). The permutation of the arguments of f goes via m = T(n, k), and this is due to sin(Pi-x) = sin(x). Of course, one can replace the product by a sum in this identity. The product identity is used in a trivial variant of Eisenstein's proof of the quadratic reciprocity law. See the W. Lang Aug 28 2016 comment under A049310. - _Wolfdieter Lang_, Aug 28 2016
%C A057059 For the proof of the (slightly extended) conjecture stated in the formula section by _L. Edson Jeffery_ see the W. Lang link. - _Wolfdieter Lang_, Sep 14 2016
%H A057059 Wolfdieter Lang, <a href="/A057059/a057059_1.pdf">Proof of a Conjecture Involving Chebyshev Polynomials.</a>
%H A057059 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%H A057059 M. P. Saclolo, <a href="http://www.ams.org/notices/201105/rtx110500682p.pdf">How a Medieval Troubadour Became a Mathematical Figure</a>, Notices Amer. Math. Soc. 58 (2011), no. 5, 682-687. See p. 684 Equation (1).
%F A057059 T(n, k) = k / 2 if k is even, n - (k - 1) / 2 if k is odd where 0 < k <= n are integers. - _Michael Somos_, Apr 21 2011
%F A057059 (Conjecture) Define the Chebyshev polynomials of the second kind by U_0(t) = 1, U_1(t) = 2*t, and U_r(t) = 2*t*U_(r-1)(t) - U_(r-2)(t) (r>1). Then T(n,k) = Sum_{j=1..n} U_(k-1)(cos((2*j-1)*Pi/(2*n+1))), 1<=k<=n. - _L. Edson Jeffery_, Jan 09 2012 (See the Sep 14 2016 comment above.)
%F A057059 From _Boris Putievskiy_, Jan 10 2013: (Start)
%F A057059 a(n) = -(A004736(n)+(A002260(n)-1)/2)*((-1)^A002260(n)-1)/2+(A002260(n)/2)*((-1)^A002260(n)+1)/2.
%F A057059 a(n) = -(j+(i-1)/2)*((-1)^i-1)/2+(i/2)*((-1)^i+1)/2, where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2). (End)
%e A057059 Formatted as a triangle T(n, k) (see Michael Somos' formula):
%e A057059 n, 2n+1\k 1 2  3 4  5 6  7 8  9 10 11 12 ..
%e A057059 1,   3:   1
%e A057059 2,   5:   2 1
%e A057059 3,   7:   3 1  2
%e A057059 4,   9:   4 1  3 2
%e A057059 5,  11:   5 1  4 2  3
%e A057059 6,  13:   6 1  5 2  4 3
%e A057059 7,  15:   7 1  6 2  5 3  4
%e A057059 8,  17:   8 1  7 2  6 3  5 4
%e A057059 9,  19:   9 1  8 2  7 3  6 4  5
%e A057059 10, 21:  10 1  9 2  8 3  7 4  6  5
%e A057059 11, 23:  11 1 10 2  9 3  8 4  7  5  6
%e A057059 12, 25:  12 1 11 2 10 3  9 4  8  5  7  6
%e A057059 ... formatted by _Wolfdieter Lang_, Aug 28 2016
%e A057059 n=4: sin identity: sin(Pi*k/9) = sin(2*Pi*T(4, k)/9), for k = 1, ..., n. That is: sin(Pi*1/9) = sin(2*Pi*4/9) = sin(Pi*(1 - 8/9)), sin(Pi*3/9) = sin(2*Pi*3/9) = sin(Pi*(1 - 6/9)). For even k this is trivial. - _Wolfdieter Lang_, Aug 28 2016
%t A057059 Table[If[OddQ@ k, n - (k - 1)/2, k/2], {n, 12}, {k, n}] // Flatten (* _Michael De Vlieger_, Aug 28 2016 *)
%o A057059 (PARI) {T(n, k) = if( k<1 || k>n, 0, if( k%2, n - (k - 1) / 2, k / 2))} /* _Michael Somos_, Apr 21 2011 */
%Y A057059 Cf. A057058, A194982; related to A141419.
%K A057059 nonn,easy
%O A057059 1,2
%A A057059 _Clark Kimberling_, Jul 30 2000